<Question>
Simplify $(u+4)(u-1) - (u-3)(u+6)$.
</Question>

<think>
<Plan_1>
Okay, so I need to simplify the expression (u + 4)(u - 1) - (u - 3)(u + 6). Hmm, let's see. I remember that to simplify such expressions, I should first expand each part using the distributive property, also known as the FOIL method for binomials, and then combine like terms. Let me take it step by step.

First, I'll expand (u + 4)(u - 1). Using FOIL:

<prompt_1.1> Expand (u + 4)(u - 1) using the FOIL method as per the above plan. </prompt_1.1>

Now, the second part is (u - 3)(u + 6). Let me expand that as well.

<prompt_1.2> Expand (u - 3)(u + 6) using the FOIL method as per the above plan. </prompt_1.2>
-----

Alternatively, maybe there's a smarter way to simplify without expanding everything? Let me think. The expression is (u+4)(u-1) - (u-3)(u+6). Maybe expanding both products and then subtracting is the straightforward way, which is what I did. There doesn't seem to be a common factor or a smarter algebraic manipulation here that would avoid expanding each term. For example, if I tried to distribute the subtraction first, but that might complicate things more.
So, expanding each product separately and then combining the terms is the way to go.
</Plan_1>

<Execution_1>
<execution_1.1>
First, multiply the First terms: u * u = u². Then the Outer terms: u * (-1) = -u. Next, the Inner terms: 4 * u = 4u. Lastly, the Last terms: 4 * (-1) = -4. So adding those up: u² - u + 4u - 4. Combine like terms: -u + 4u is 3u, so the first part becomes u² + 3u - 4.
</execution_1.1>

<execution_1.2>
First, the u terms: u * u = u². 

Then the outer terms: u * 6 = 6u. 

Next, the inner terms: -3 * u = -3u. 

Lastly, the last terms: -3 * 6 = -18. 

Combine those: u² + 6u - 3u - 18. Combine like terms: 6u - 3u is 3u. So the second part becomes u² + 3u - 18.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2:
Now, putting it all back into the original expression: (u² + 3u - 4) - (u² + 3u - 18). Wait, I need to subtract the entire second expanded expression. So I have to distribute the negative sign to each term inside the parentheses.
Let's do that carefully.

<prompt_2.1> Subtract the expanded form of (u - 3)(u + 6) from the expanded form of (u + 4)(u - 1) as per the above plan. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
First, write down the first expanded expression: u² + 3u - 4. Then subtract the second expanded expression: -(u²) - 3u + 18. Because subtracting each term: -u² (since it's -1*u²), -3u (from -6u - (-3u? )). Wait, hold on. Wait, the second expansion was u² + 3u - 18. So subtracting all of that would be -u² - 3u + 18. Let me check that again. If the second part is u² + 3u - 18, then subtracting it would be -u² - 3u + 18. Yes, that's correct.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Now, combine the two parts: u² + 3u - 4 - u² - 3u + 18. Let's combine like terms.

<prompt_3.1> Combine the terms from both expansions to simplify the expression as per the above plan. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
u² - u² cancels out. Then 3u - 3u cancels out as well. Then, -4 + 18. That leaves 14. Wait, so everything cancels except for 14?
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Let me check my steps again to make sure I didn't make a mistake.

<prompt_4.1> Verify the steps of expanding and simplifying as per the above plan. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
First expansion: (u + 4)(u - 1). FOIL gives u² - u + 4u - 4 = u² + 3u - 4. That seems right. Second expansion: (u - 3)(u + 6). FOIL gives u² + 6u - 3u - 18 = u² + 3u - 18. Correct. Then subtracting the second from the first: (u² + 3u - 4) - (u² + 3u - 18). Distribute the negative sign: u² + 3u - 4 - u² - 3u + 18. Combine terms: u² - u² = 0, 3u - 3u = 0, -4 + 18 = 14.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Hmm. So the simplified form is 14? That seems surprisingly simple. Let me verify by plugging in a value for u in both the original expression and the simplified result to see if they match.

<prompt_5.1> Verify the simplified expression by substituting values for u as per the above plan. </prompt_5.1>

Wait, maybe I made a mistake in the signs when distributing the negative sign. Let me check that again.

<prompt_5.2> Recheck the distribution of the negative sign when subtracting the expanded form as per the above plan. </prompt_5.2>
</Plan_5>

<Execution_5>
<execution_5.1>
Let's take u = 2.

Original expression: (2 + 4)(2 - 1) - (2 - 3)(2 + 6) = (6)(1) - (-1)(8) = 6 - (-8) = 6 + 8 = 14. 

Simplified result is 14. Okay, that matches. Let's try another value, say u = 0.

Original expression: (0 + 4)(0 - 1) - (0 - 3)(0 + 6) = (4)(-1) - (-3)(6) = -4 - (-18) = -4 + 18 = 14. 

Again, 14. Hmm, interesting. So regardless of the value of u, the expression simplifies to 14.
</execution_5.1>

<execution_5.2>
Original expression after expanding both parts: (u² + 3u - 4) - (u² + 3u - 18). So that's u² + 3u -4 - u² -3u +18. Yes, because subtracting each term in the second polynomial. So u² - u² = 0, 3u -3u = 0, -4 +18 = 14.
</execution_5.2>
</Execution_5>

<Final_answer>
Therefore, the simplified form of the given expression is $\boxed{14}$.
</Final_answer>
</think>